## multilevel linear models, Gibbs samplers, and multigrid decompositions

**A** paper by Giacommo Zanella (formerly Warwick) and Gareth Roberts (Warwick) is about to appear in Bayesian Analysis and (still) open for discussion. It examines in great details the convergence properties of several Gibbs versions of the same hierarchical posterior for an ANOVA type linear model. Although this may sound like an old-timer opinion, I find it good to have Gibbs sampling back on track! And to have further attention to diagnose convergence! Also, even after all these years (!), it is always a surprise for me to (re-)realise that different versions of Gibbs samplings may hugely differ in convergence properties.

At first, intuitively, I thought the options (1,0) (c) and (0,1) (d) should be similarly performing. But one is “more” hierarchical than the other. While the results exhibiting a theoretical ordering of these choices are impressive, I would suggest pursuing an random exploration of the various parameterisations in order to handle cases where an analytical ordering proves impossible. It would most likely produce a superior performance, as hinted at by Figure 4. (This alternative happens to be briefly mentioned in the Conclusion section.) The notion of choosing the optimal parameterisation at each step is indeed somewhat unrealistic in that the optimality zones exhibited in Figure 4 are unknown in a more general model than the Gaussian ANOVA model. Especially with a high number of parameters, parameterisations, and recombinations in the model (Section 7).

An idle question is about the extension to a more general hierarchical model where recentring is not feasible because of the non-linear nature of the parameters. Even though Gaussianity may not be such a restriction in that other exponential (if artificial) families keeping the ANOVA structure should work as well.

Theorem 1 is quite impressive and wide ranging. It also reminded (old) me of the interleaving properties and data augmentation versions of the early-day Gibbs. More to the point and to the current era, it offers more possibilities for coupling, parallelism, and increasing convergence. And for fighting dimension curses.

“in this context, imposing identifiability always improves the convergence properties of the Gibbs Sampler”

Another idle thought of mine is to wonder whether or not there is a limited number of reparameterisations. I think that by creating unidentifiable decompositions of (some) parameters, eg, μ=μ¹+μ²+.., one can unrestrictedly multiply the number of parameterisations. Instead of imposing hard identifiability constraints as in Section 4.2, my intuition was that this de-identification would increase the mixing behaviour but this somewhat clashes with the above (rigorous) statement from the authors. So I am proven wrong there!

Unless I missed something, I also wonder at different possible implementations of HMC depending on different parameterisations and whether or not the impact of parameterisation has been studied for HMC. (Which may be linked with Remark 2?)

October 22, 2021 at 2:59 am

I discuss the parameterization considerations for Hamiltonian Monte Carlo in https://betanalpha.github.io/assets/case_studies/hierarchical_modeling.html and https://betanalpha.github.io/assets/case_studies/factor_modeling.html.